2012/08/03 by Bolton, John, Dillen, Franki, Dioos, Bart +1
#53C40 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1208.0737
In this paper almost complex surfaces of the nearly Kähler S3× S3 are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kähler S3× S3. We also find a correspondence between almost complex surfaces in the nearly Kähler S3× S3 and solutions of the general H-system equation introduced by Wente, thus obtaining a geometric interpretation of solutions of the general H-system equation. From this we deduce a correspondence between constant mean curvature surfaces in \mathbb R3 and almost complex surfaces in the nearly Kähler S3× S3 with vanishing holomorphic differential. This correspondence allows us to obtain a classification of the totally geodesic almost complex surfaces. Moreover, we will prove that almost complex topological 2-spheres in S3× S3 are totally geodesic. Finally, we also show that every almost complex surface with parallel second fundamental form is totally geodesic.